The Mind of a Theorist
The Return

by George E. Hrabovsky, President of MAST

The News

Well, hello again, this time in a new format. I have to admit that I have never written a piece for electronic-only dissemenation. It is good to be contributing to SAS again. My columns will be, from now on (assuming you all like the new content) in three parts: I will have a brief news piece (either MAST news that bears on theory, and announcement of some new and important theoretical breakthrough, or something that triggers a response from general news), some demonstration of theory in action (similar to the lab notes that have been appearing), and a book review related to the demonstration.

I will be writing this column in Mathematica and then converting it to HTML for publication in the Bulletin and to my web site, www.madscitech.org. Simply look for the link to The Mind of a Theorist.

Demonstration: Proving Something to be a Group

Some time ago I wrote a column on symmetry and why this fascinating subject is important. The bottom line was that if something is symmetrical then there is some reason to believe that there is a conservation law at work somewhere. How do we find whether something is symmetrical? One way is to show that it forms a symmetry group. I will explore this special type of group in my next column. In this column I will show you how to prove something is a group, this will lay the ground-work for my next column.

As you no doubt know by now any collection of things is naively a set. The elements of such sets can be combined two at a time, an example of this is addition. Such combinations are called binary relations. Any set whose elements are combined by a binary relation is called an algebraic structure. We will use the symbol ⊕ to denote such relations. Let us use addition as our binary relation. We can now choose what sets we want to work with. Let us choose two sets; the set of natural numbers,

and the set of integers,

The simplest type of algebraic structure has two special properties for its binary relations. The first is the requirement that any results from the relation must also be members of the set, this is called closure. We can write this symbolically using x and y as elements of the set S,

We can immediately apply this to our two example sets. We simply ask the question, "If we add two elements of these sets, do we get another element of these sets?" In both cases the answer is clearly yes.

The second requirement is that when we apply the binary relation to three elements, it does not matter which pair we relate first, this property is called associativity. Symbolically this can be written,

We can immediately apply this to our two example sets. We simply ask the question, "Is addition associative on these sets?" In both cases the answer is  yes. This is well established from our study of elementary algebra in high school. Such an algebraic structure is called a semigroup. Under addition we can say that both the set of natural numbers and the set of integers form a semigroup.

A third requirement can be added to our semigroup. If we state that there must be an element such that the application of our binary relation between that element and any other element of the set leave that other element unchanged, we have established an idenity element. This can be written symbolically as,

Any semigroup meeting this requirement is called a monoid. We can apply this to our two semigroups, "Do these semigroups have an identity element?" We recall that the identity element for addition is 0. The set of natural numbers does not have this element, though the set of integers does. What conclusion can we reach from this? The set of natural numbers is not a monoid, while the set of integers is.

If we have a monoid, we can check for another requirement. If we state that there must be an element such that the application of our binary relation between that element and any other element of the set gives us the identity element, we have established an inverse element. This can be written symbolically as,

Any monoid meeting this requirement is called a group. We can apply this to our monoid, "Does the monoid of the set of integers under addition have an inverse element?" The answer is yes, thus the set of integers is a group under addition.

In the next column I will present the idea of a symmetry group.

Book Review

The subject of groups is a part of the discipline called abstract algebra. It is a generalization of the basic notions of algebra we went through in high school. The best book that is accessible to people is the subject of this review:

Algebra, 3rd Edition, Saunders MacLane and Garrett Birkhoff, Published in 1993 by AMS-Chelsea.
This book begins with a detailed study of the algebra of sets, functions, and numbers. Then it builds on these ideas to study such algebraic structures as groups, rings, vector spaces, and many more. This is my favorite book on algebra. It has LOTS of practice problems in a nice balance between actual calculations and proofs. Be prepared to work hard. If you do put the work in, it is immensely rewarding and you will come away with a deep understanding of these concepts. All aspectsof algebra are being applied to physics, so this subject area is a must for theorists.